As is known, currently there is a great deal of research activity in the field of optics, based on photons with different OAM states (or modes).
In particular, researchers have discovered that photons can carry both a Spin Angular Momentum (SAM) associated with polarization, and an OAM associated with the phase azimuthal profile orthogonal to the propagation axis.
In detail, in optics, OAM is a component of angular momentum of a light beam that depends on the field spatial distribution, and not on the polarization of the light beam. An example of OAM is the OAM appearing when a paraxial light beam is in a so-called helical (or twisted) mode, namely when its wavefront is shaped as a helix with an optical vortex in the center, at the beam axis. The helical modes are characterized by an integer number m, positive or negative. Said integer m is also called “topological charge” of the optical vortex.
In this connection, FIG. 1 schematically illustrates exemplary OAM modes of a light beam.
In particular, the OAM modes shown in FIG. 1 correspond to five different topological charges, specifically to m=0, ±1, and ±2.
In detail, FIG. 1 shows:                in a first column, beam wavefront shapes for m=0, ±1, ±2;        in a second column, optical phase distributions in beam cross-sections for m=0, ±1, ±2; and,        in a third column, light intensity distributions in beam cross-sections for m=0, ±1, ±2.        
As shown in FIG. 1, if m=0, the mode is not helical and the wavefronts are multiple disconnected surfaces, in particular a sequence of parallel planes (from which the name “plane wave”). If |m|≧1, the wavefront is helical with handedness determined by the sign of in, and the beam photons have an OAM state (or mode) of ±mh directed along the beam axis, where h denotes the Dirac constant which, as known, is obtained as
      ℏ    =          h              2        ⁢                                  ⁢        π              ,where h denotes the Planck constant.
Different OAM states are mutually orthogonal thereby allowing, in principle, to transmit any number of bits per photon and, thence, to increase transmission capacity.
In consideration of OAM potentialities of increasing transmission capacity and as RF spectrum shortage problem is deeply felt in radio communications sector, recently a lot of experimental studies have been carried out on the use of OAM modes also at RF in order to try to enhance RF spectrum reuse.
In this connection, reference may, for example, be made to:                Mohammadi S. M. et al., “Orbital Angular Momentum in Radio—A System Study”, IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE SERVICE CENTER, PISCATAWAY, NJ, US, vol. 58, no. 2, 1 Feb. 2010, pages 565-572, which shows that standard antennas arranged in circular arrays can be used to generate RF beams carrying OAM;        Tamburini F. et al., “Encoding many channels in the same frequency through radio Vorticity: first experimental test”, arXiv.org, 12 Jul. 2011, Ithaca, N.Y., USA, which experimentally shows that it is possible to propagate and use the properties of twisted non-monochromatic incoherent radio waves to simultaneously transmit several radio channels on one and the same frequency by encoding them in different (and, thence, orthogonal) OAM states (even without using polarization or dense coding techniques); and        GB 2 410 130 A, which discloses a planar phased array antenna for transmitting and receiving OAM radio vortex modes, which antenna comprises a circular array of cavity backed axial mode spiral antenna elements whose phase is controlled such that the phase of each antenna element changes sequentially about the array.        
From a mathematical perspective, the transmission of an OAM mode (or state) at a single RF (i.e., by using a pure tone) implies that the electrical field on the radiating aperture can be represented as:F(ρ,φ)=F(ρ)ejkφ,where ρ and φ are the cylindrical coordinates on the radiating aperture, and k is a positive or negative integer number.
The radiated field can be represented in the far zone as:
            E      ⁡              (                  ϑ          ,          φ                )              =                  1        R            ⁢                        ∫          ∫                S            ⁢              F        ⁡                  (                      ρ            ,            ϕ                    )                    ⁢              ⅇ                              -            j                    ⁢                                          ⁢          2          ⁢                                          ⁢          π          ⁢                      ρ            λ                    ⁢          si          ⁢                                          ⁢                      n            ⁡                          (              ϑ              )                                ⁢          co          ⁢                                          ⁢                      s            ⁡                          (                              φ                -                ϕ                            )                                          ⁢      ρ      ⁢              ⅆ        ρ            ⁢              ⅆ        ϕ              ,where 9 and φ are the spherical coordinates in the far field, R denotes the radius of the sphere centered on the radiating aperture, S denotes the integration surface used on reception side, and λ denotes the wavelength used.
As is known, due to intrinsic characteristics of OAM, an OAM mode transmitted at a single RF (i.e., by using a pure tone) is affected by a phase singularity which creates a null at the bore-sight direction, thereby resulting thatE(0,0)=0.
In order for said phase singularity to be compensated, the integration surface S used on reception side should be sized so as to include the crown peak generated by the OAM mode.
In particular, the integration surface S used on reception side should be different for each OAM mode and, considering the sampling theorem applied to the radiating antenna, should have an area given by:
            Δ      ⁢                          ⁢      S        =                  Δ        ⁢                                  ⁢        Ω        ⁢                                  ⁢                  R          2                    =              2        ⁢                              (                                          λ                D                            ⁢              R                        )                    2                      ,where D denotes the diameter of the radiating antenna.
Therefore, the price to be paid with pure OAM modes transmitted by using pure tones (i.e., single radiofrequencies) is that the dimensions of the equivalent receiving antenna depend on the distance R from, and on the diameter D of, the transmitting antenna. For example, for a transmitting antenna having a diameter D of about 1 m and working at 2.4 GHz, a receiving antenna located at a distance R of about 400 m should be a ring with a diameter of about 55 m. As shown in FIG. 2, by reducing the diameter of the ring-shaped receiving antenna an additional loss of the order of 12 dB at 5.5 m appears.
This solution is impractical for satellite communications, where the aperture efficiency and the size of the antennas are very critical issues. For example, in geostationary-satellite-based communications in Ka band, for a ground antenna having a diameter D of about 9 m, the diameter of the receiving ring on board the geostationary satellite should be of the order of 50 Km, thereby resulting impractical.
Thence, in view of the foregoing, the main criticality in using radio vorticity in practical systems is that the orthogonality between OAM modes depends on the size of antennas, on the distance between the transmitting and receiving antennas, and on the need for the receiving antenna to operate as an interferometer basis (as, for example, disclosed in the aforesaid papers “Orbital Angular Momentum in Radio—A System Study” and “Encoding many channels in the same frequency through radio Vorticity: first experimental test”, and in GB 2 410 130 A). The result of these constraints is that currently known OAM-based radio communication systems are inefficient and unusable for very long distances such as the ones involved in satellite communications.